

\chapter{二阶常微分方程的级数解法~~本征值问题}

\section{习题1-5}

\begin{yyEx}
	在$x = 0$的邻域内, 求解方程$(1-x^2)y''+xy'-y = 0$.
\end{yyEx}

\begin{yyEx}
	在$x = 0$的邻域内求解方程$y'' + y = 0$.
\end{yyEx}

\begin{yyEx}
	在$x = 0$的邻域内求解艾里方程$y'' - xy = 0$.
\end{yyEx}

\begin{yyEx}
	求方程$x^2y''(x)-xy'(x)+y = 0$在$x = 0$的邻域内的通解.
\end{yyEx}

\begin{yyEx}
	将下列方程化为施图姆-刘维尔型方程的标准形式:
	\begin{equation*}
		(1)y''-\cot xy'+\lambda y = 0;~~(2) xy''+(1-x)y'+\lambda y = 0.
	\end{equation*}
\end{yyEx}

\section{习题6-10}

\begin{yyEx}
	求解下列本征值问题的本征值和本征函数:
	\begin{enumerate}
		\item $X''(x) + \lambda X(x) = 0, X(0) = 0 , X'(l) = 0$;\\
		\item $X''(x) + \lambda X(x) = 0, X'(0) = 0 , X(l) = 0$;\\
		\item $X''(x) + \lambda X(x) = 0, X(0)+HX'(0) = 0 , X(l) = 0$(H\text{为常数});\\
		\item $\begin{lgathered}
			\frac{\mathrm{d}}{r\mathrm{d}r}\left( r\frac{\mathrm{d}R}{\mathrm{d}r} \right)+\frac{\lambda}{r^2}R = 0,R(a) = 0,R(b) = 0,~~0<a<b.
		\end{lgathered}$
	\end{enumerate}
\end{yyEx}

\begin{yyEx}
	已知二阶线性常微分方程的两个线性无关解$y_1(x) = e^{a/x}$和$y_2(x) = e^{-a/x}$, 求其所满足的方程.
\end{yyEx}

\begin{yyEx}
	在$x = 0$的邻域内求解方程$y'' - 2xy' + (\lambda-1)y = 0$, 当$\lambda$取什么数值时可使级数退化为多项式.
\end{yyEx}

\begin{yyEx}
	求合流超几何方程$xy''(x)+(\gamma -x)y'(x) - \alpha y(x) = 0$在$x = 0$附近的通解, 其中$\alpha,\gamma$为常数, 且$\alpha>0,1-\gamma\neq 0,1,2,\cdots$.
\end{yyEx}

\begin{yyEx}
	证明在下列有界条件下的本征值问题中, 本征函数是正交的.
	\begin{equation*}
		\begin{dcases}
			&\frac{\mathrm{d}}{\mathrm{d}x}\left[ k(x)\frac{\mathrm{d}y(x)}{\mathrm{d}x} \right] + \lambda y(x) = 0,~~x\in(a,b), \\
			&\abs{y(a)}<+\infty,~~\abs{y(b)}<+\infty,
		\end{dcases}
	\end{equation*}
	其中$k(x)$为非负的连续实函数, 且$k(a) = k(b) = 0$.
\end{yyEx}